Math Camp 2020: Exercise Set #1

1. For each of the following statements, determine whether the statement is true or false, and if

false indicate why it’s false.

(a) {(2, 1, 1), (2, 1), (1, 1)} ⊆ R3 × R2.

(b) {(2, 1, 1), (2, 1), (1, 1)} ⊆ R3 ∪ R2.

(c) {(2, 1, 1), (2, 1), (1, 1)} ⊆ R3 × R2 × R2.

(d)((2, 1, 1), (2, 1), (1, 1)

)∈ R3 × R2 × R2.

(e){(

(2, 1, 1), (2, 1), (1, 1))}⊆ R3 × R2 × R2.

(f) ∃n ∈ N : {(2, 1, 1), (2, 1), (1, 1)} ⊆ Rn.

(g) (1, 2, 3) ⊆ N.

(h) (1, 2, 3) ∈ N.

(i) (1, 2, 3) ∈ R3.

(j) {(2, 1, 1), (2, 1), (1, 1)} = {(2, 1), (2, 1, 1), (1, 1)}.

(k) R2 ∪ R3 = R3 ∪ R2.

(l) R2 × R3 = R3 × R2.

2. In our first Logic lecture (Lecture #5) we listed compound statements that covered 11 of the 16

possible “profiles” of truth values. Using only negation, conjunction, and disjunction, for each of

the remaining five truth-value profiles construct a compound statement that has that truth-value

profile.

3. Let A be an m× n matrix and let f be the function f(x) = Ax. Sections 7.3 and 7.4 of Simon& Blume might be helpful for this exercise. Drawing diagrams will also be helpful.

(a) What is the domain of f and what is the target space of f?

(b) What conditions on m,n, and A are both necessary and sufficient for f to be one-to-one and

onto its target space?

(c) If m = 2 and n = 1, under what conditions (if any) will f be one-to-one? Onto?

(d) If m = 2 and n = 3, under what conditions (if any) will f be one-to-one? Onto?

4. “You can fool some of the people all the time, and you can fool all the people some of the

time, but you can’t fool all the people all the time.” (Attributed by some to Abraham Lincoln.)

Hammack’s Exercise #2.9.12 asks you to translate this sentence into symbolic logic. But the

sentence is ambiguous — i.e., it can be translated into different formal statements that have

different truth values.

Let’s define the following three sentences, or statements:

SA: You can fool some of the people all the time.

AS: You can fool all the people some of the time.

AA: You can fool all the people all the time.

It’s clear that the original sentence is the conjunction SA ∧ AS ∧ ∼ AA. However, the sentencesSA and AS, as written here in English, are ambiguous — as you’re going to demonstrate.

Let’s say there are three people (Abby, Beth, and Carl) and three times (Monday, Wednesday, and

Friday). Let X = {a, b, c} and Y = {m,w, f}. Let’s write the (open) statement “x can be fooledat time y” as ϕ(x, y). Note that you can think of ϕ as a function, ϕ : X × Y → {T, F}, where Tor F is the truth value of ϕ(x, y). You can also write Φ = {(x, y) |ϕ(x, y) = T}, the set of pairs(x, y) for which x can be fooled at time y. Figure 1 depicts the statement AA; Figure 2 depicts the

statement “Abby can be fooled on Monday and Wednesday (only), Beth can be fooled on Monday

(only), and Carl can be fooled on Monday and Friday (only).”

Now note that the statements SA and AS can each have two different meanings:

SA could mean ∃x ∈ X: ∀y ∈ Y : ϕ(x, y) = T , which we’ll denote as SA1, orSA could mean ∀y ∈ Y : ∃x ∈ X: ϕ(x, y) = T , which we’ll denote as SA2.

AS could mean ∃y ∈ Y : ∀x ∈ X: ϕ(x, y) = T , which we’ll denote as AS1, orAS could mean ∀x ∈ X: ∃y ∈ Y : ϕ(x, y) = T , which we’ll denote as AS2.

As in Figures 1 and 2, for each of the following four statements Li (i = 1, 2, 3, 4) draw a diagram

of the Cartesian product X × Y , and use the diagram to represent a function ϕ (i.e., a set Φ) forwhich the statement is True and as many of the other three statements as possible are False. (For

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each statement Li there are multiple correct solutions — i.e., multiple correct sets Φ ⊆ X × Y .You only need to draw one correct solution for each Li.) In each of the four cases indicate which

statements are True and which are False.

L1 = SA1 ∧ AS1 ∧ ∼ AA L3 = SA2 ∧ AS1 ∧ ∼ AAL2 = SA1 ∧ AS2 ∧ ∼ AA L4 = SA2 ∧ AS2 ∧ ∼ AA.

5. Now let X and Y be arbitrary sets and let ϕ(x, y) be an arbitrary open statement (or Φ an

arbitrary subset of X × Y ). Prove that SA1 implies SA2 or that AS1 implies AS2. (These twoimplications are clearly equivalent to one another, so you only need to prove one of them.) This

proof should require no more than a sentence or two.

Note: Understanding Exercises #4 and #5 will pretty much guarantee that you will have no trou-

ble understanding uniform continuity and uniform convergence. Failing to understand these two

exercises will likely mean you’ll have a hard time with uniform continuity and uniform convergence.

6. Let a ∈ R and n ∈ N, and define the function f : R→ R by f(x) = axn.

(a) Use your knowledge of proof by induction to verify that if n is even, then f is neither one-to-one

nor onto R.

(b) Assuming that the function f is continuous (it is, but don’t prove it here), can you determine

whether f is one-to-one and/or onto if n is odd instead of even? You might want to use the

Intermediate Value Theorem here. (Oddly, the Intermediate Value Theorem doesn’t appear in

Simon & Blume. It’s on page 60 of Sundaram, and a Google search will also work fine; for

example, https : //en.wikipedia.org/wiki/Intermediate value theorem.

7. (Hammack, Chapter 10, Exercise #2)

Prove that for every n ∈ N, 12 + 22 + 32 + · · ·+ n2 = n(n + 1)(2n + 1)6

.

8. For each of the following vectors, determine the vector’s length and determine a vector of length

1 that points in the opposite direction.

(a) (12, 5) (b) (1, 1) (c) (3, 3) (d) (3,−3) (e) (−1, 1,−1)(f) (1, 1, 1, 1) (g) (12, 0, 0, 5) (h) (12,−1, 1, 5).

9. For each of the following vectors v, find two vectors u and w that are orthogonal to v and have

the same length as v:

(a) v = (1, 0) (b) v = (1, 1) (c) v = (1, 0, 0) (d) v = (1, 1, 1).

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10. Draw a diagram depicting the following, in R2: the vector a = (1,−1); the lineH = {x ∈ R2 | a · x = 0}; and the vectors (1, 1), (1, 0), (2, 1), (0,−1), (0, 1), (−1, 2), (−1, 0), (−1, 1).Denoting each of these vectors in turn by v, determine a · v; which side of the line H the vector vlies on (lower-right (SE) or upper-left (NW)); and, both visually (by looking at the diagram) and

analytically (via the dot product) whether the angle between a and v is acute, obtuse, or a right

angle (they’re orthogonal).

Here’s something I find helpful for visualizing things in R3: Assuming you’re in a room with wallsthat are perpendicular to one another and to the floor, look at a corner where two of the walls meet

the floor; imagine the positive x1-axis to be the line where the wall to your left meets the floor;

the positive x2-axis to be the line where the wall to your right meets the floor; and the positive

x3-axis to be the line where the two walls meet.

11. Let a be the vector a = (1, 1, 1) and let H be the plane H = {x ∈ R3 | a · x = 0} in R3.For each of the following vectors v determine analytically (via the dot product) whether the angle

between a and v is acute, obtuse, or is a right angle (they’re orthogonal); and whether v lies on

the same side of H as the vector a, on the opposite side of H, or on the plane H:

(a) v = (1, 0, 0) (b) v = (0, 0, 1) (c) v = (0,−1, 0) (d) v = (0, 1, 1) (e) v = (1,−1, 0).

12. Prove that if n > 1 then the x1-axis in Rn is a closed set (or equivalently, prove that itscomplement is an open set).

13. Prove that every finite subset of Rn is a closed set.

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Math Camp 2020: Exercise Set #2

1. The set R∞ of sequences of real numbers is a vector space under the usual (component-wise)definitions of vector addition and scalar multiplication:

(x1, x2, . . .) + (y1, y2, . . .) := (x1 + y1, x2 + y2, . . .) and α(x1, x2, . . .) := (αx1, αx2, . . .).

Verify that the set `∞ of bounded sequences of real numbers, with the operations of vector addition

and scalar multiplication it inherits from R∞, is a vector subspace of R∞. [A sequence (xn)∞1 ∈ R∞

is bounded if ∃M ∈ R : ∀n ∈ N : |xn| 5M . See this link for an application to economics.]

2. Let S be the set of all real sequences that have only a finite number of non-zero terms — i.e.,

S ={

(xn)∞1 ∈ R∞ | {n ∈ N | xn 6= 0} is a finite set

}. Is S a vector subspace of R∞? Is S a

vector subspace of `∞? Verify your answers.

3. Provide a proof of the following proposition: If dimV = n and v1, . . . ,vn span V , then

{v1, . . . ,vn} is a basis of V .

4. At about 32:00 of Lecture 14(B) I say“You might guess that R∞ is infinite-dimensional. Well,let’s see.” Then I go on to instead check whether the infinite set consisting of the unit vectors is

a basis. In the end I point out that we didn’t check whether R∞ has a finite basis, and that thatwould be a good exercise. Here’s that exercise. It’s a little bit challenging, but what we’re going

to do is a good example of how to prove something when it’s not clear how to do it. The original

problem — to show that no finite set can be a basis of R∞ — is part (d), and that was originallygoing to be this exercise. When I tried to do it myself I came up with basically the right idea

pretty quickly, but I couldn’t see exactly how to work out the details. So what I did is what one

typically has to do in this situation: you cook up a simpler version of the problem and solve that,

hoping that that will give you the insight how to solve the real problem. What we’re going to do

here is go several steps simpler — working backward from (d), getting simpler at each step — and

work our way back up to the actual problem. (It’s still going to be kind of challenging.)

(a) Prove that the set {a1, a2}, in which a1 = (1, 2, 1) and a2 = (2, 1, 1), is not a basis of R3.

(b) Prove that no subset of R3 with only two vectors can be a basis of R3.

(c) Prove that the set {a1, a2}, in which a1 = (1, 2, 1, 1, 1, . . .) and a2 = (2, 1, 1, 1, 1, . . .) is not abasis of R∞.

(d) Prove that no subset of R∞ with only a finite number of vectors can be a basis of R∞.

https://en.wikipedia.org/wiki/L-infinity

5. Let f : R3 → R3 be the linear function for which f(1, 0, 0) = (1, 1, 1), f(0, 1, 0) = (0, 1, 0), andf(0, 0, 1) = (2, 3, 2).

(a) Write down the matrix A that defines f — i.e., the matrix for which f(x) = Ax for all x ∈ R3.

The column space of A is the space spanned by the columns of A; the row space of A is the space

spanned by the rows of A; and the nullspace of A is the set of solutions of the equation system

Ax = 0 — denoted Col(A), Row(A), and Null(A), respectively.

(b) Determine bases for each of the spaces Col(A), Row(A), and Null(A).

(c) Is f one-to-one? Is it onto R3? Explain how you determined the answers to these two questions.

6. Provide a proof by induction that if S is a convex set in a vector space V , then every convex

combination of vectors in S is also in S. Note that this, together with its converse, is a theorem

in Lecture 15. As we said there, the converse is a trivial immediate consequence of the definition

of a convex set. The initial step of the induction proof here is equally trivial: that every convex

combination of two vectors in S is also in S, which is obviously true, as the definition of a convex

set.

7. Exercise #21.19 in Simon & Blume. Note that the definition of quasiconcave function they’re

using is the one that says a quasiconcave function is one for which all upper-contour sets are

convex.

8. Here’s a useful theorem: If f1 and f2 are strictly concave functions defined on a convex subset

of a vector space V , then the sum f1 + f2 is also strictly concave.

(a) Provide a proof of the theorem.

(b) Let u(x1, x2) =√x1 +

√x2 be a utility function defined on R2+. The real function f(z) =

√z

is strictly concave. Explain why the theorem doesn’t apply to this situation — the theorem can’t

be invoked to establish that u is strictly concave.

Here’s another extremely useful theorem: If X1 and X2 are intervals in R and if f1 : X1 → Rand f2 : X2 → R are strictly concave functions, then the function f : X1 × X2 → R defined byf(x1, x2) = f1(x1)+f2(x2) is strictly concave. Note that this theorem does imply that the function

u(x1, x2) =√x1+√x2 is strictly concave (because the real function f(z) =

√z is strictly concave).

(c) Provide a proof of this second theorem.

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Math Camp 2020: Exercise Set #3

1. Exercise #10.16 in Simon & Blume. Note how much easier it is to prove the Triangle Inequality

for these norms than it is for the Euclidean norm, where we needed to first prove the Cauchy-

Schwarz Inequality. This is one more example of how one or the other of these two norms is often

easier to work with than the Euclidean norm. [In the S&B exercise the norms are written with

three vertical lines on each side. That’s a typo; they should be just ‖(u1, u2)‖.]

2. Prove that if a sequence {x(n)} converges to x ∈ R`, then for each k ∈ {1, . . . , `} the componentsequence {xk(n)} converges to xk. [In Lecture 18 we said that if a set in R` is open in one norm onR` then it’s open in all other norms on R`, so you can use whichever one of the norms ‖ · ‖1, ‖ · ‖2,or ‖ · ‖∞ that makes your proof easiest.]

3. Our definition of a compact set is a set in which every sequence has a subsequence that converges

to a point in the set. Prove directly from this definition that every finite set of points in Rn iscompact. [As in #1, you can use any norm on Rn here.]

4. Determine whether the utility function u : R2+ → R is continuous if u is defined by

u(x1, x2) =

x1x2, if x1x2 ≤ 4

4, if 4 < x1x2 < 9

x1x2 − 5, if x1x2 ≥ 9

5. There are two elements of “doing proofs”. One element is being able to create a proof of

something. The second element, which is necessary for the first one, is being able to tell whether a

proof is valid — i.e., whether a given proof actually succeeds at establishing the proposition we’re

trying to prove. Each of the following exercises consists of a conjecture and a proposed proof. In

each case the conjecture may be true or false, and the proof may be correct (valid) or not. Of

course, we can’t have a conjecture that’s false and a proof that’s valid. What you’re to do, for

each exercise, is to determine which of the three remaining categories the exercise falls into: (a)

the conjecture is false (in this case, provide a counterexample to verify that it’s false and point out

why the proof is not valid); (b) the conjecture is true and the proof is valid (here you only need

to say that’s the case); or (c) the conjecture is true but the proof is not valid (in this case, point

out why the proof is invalid and fix it or otherwise give a correct proof).

Conjecture 1: Let X ′ = [0, 1], the closed unit interval in R, with the usual norm (the absolutevalue). The set S = [0, 1

2) = {x ∈ X ′ | 0 5 x < 1

2} is open in X ′ (i.e., open relative to X ′).

Proof?: Let V = (−12, 12), the open interval in R between −1

2and 1

2. Then S = V ∩X ′ and

V is open in R, so S is open in X ′. �

Conjecture 2: Let {x(n)} be a sequence in R`. If for each k ∈ {1, . . . , `} the component sequence{xk(n)} converges to xk ∈ R, then {x(n)} converges to x = (x1, . . . , x`) ∈ R`. [Note that this isthe converse of a proposition you proved in a previous exercise.]

Proof?: Let � > 0. Then for every k ∈ {1, . . . , `} there is an nk ∈ N such that n > nk ⇒|xk(n) − xk| < 1` �. Therefore

∑`k=1 |xk(n) − xk| < �; i.e., ‖x(n) − x‖1 < �. Since this is true for

every � > 0, {x(n)} converges to x. � [Do you think ‖ · ‖1 was the best norm to use?]

Conjecture 3: The set S = {x ∈ Q | 0 5 x <√

2 } — i.e., the set of nonnegative rationalnumbers less than

√2 — is not closed in Q.

Proof?: Let {xn} be a sequence of rational numbers in S (each is therefore less than√

2)

that converges to√

2. Since√

2 /∈ S, S is not a closed set. �

Conjecture 4: If {xn} converges to x in R`, then every subsequence of {xn} converges to x.

Proof?: Let {xnm} be a subsequence of {xn} and suppose {xnm} converges to a point x̃.We’ll show that x̃ = x. Suppose instead that x̃ 6= x, and we’ll obtain a contradiction. Let� = 1

2‖x̃− x‖. Then

∃m ∈ N : m > m⇒ ‖xnm − x̃‖ < � and ∃n ∈ N : n > n⇒ ‖xn − x‖ < �.

Let m satisfy m > m and nm > n. Then ‖xnm− x̃‖ < � and ‖xnm−x‖ < �. Therefore the TriangleInequality yields ‖x̃− x‖ 5 ‖xnm − x̃‖+ ‖xnm − x‖ < 2�, a contradiction, since ‖x̃− x‖ = 2�. �

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Math Camp 2020: Exercise Set #4

1. Determine whether each of the following matrices is positive definite or semidefinite, or is

indefinite. If the matrix is indefinite, provide a vector at which the associated quadratic form is

positive and one at which the quadratic form is negative. If the matrix is semidefinite, provide a

vector x 6= 0 at which the quadratic form has the value zero.

(a)

1 0 2

0 3 4

2 4 0

(b)

1 0 2

0 3 3

2 3 7

(c)−1 0 −2

0 0 0

−2 0 −1

(d)−3 0 2

0 −1 02 0 −3

2. Use the differential function (also called the total differential, or the total derivative, as Simon

& Blume call it) to give an informal argument why the gradient of a function f : R2 → R at x ∈ R2

is orthogonal to the function’s level curve at x. Then recast the argument so that it applies to

functions f : Rn → R.

3. The function

f(x) =

{x2, x 5 1

3x− 2, x = 1

is continuous everywhere in R. (Note that f(1) = 1.) By direct appeal to the definition of thederivative, show that f is not differentiable at x = 1.

4. A C2 function f : X → R, defined on an open convex set X ⊆ Rn, is concave if and only ifits 2nd-degree Taylor polynomial is concave at every x ∈ X, which is the case if and only if theHessian matrix of second partial derivatives, D2f(x), or H(x), is negative semidefinite at every

x ∈ X. Moreover, a sufficient condition for f to be strictly concave is that the Hessian matrix benegative definite at every x ∈ X.

(a) Provide a counterexample to show that a negative definite Hessian matrix is not necessary for

a C2 function to be strictly concave.

(b) For parameter values a1, a2, and a3 define the function f : R2++ → R by f(x1, x2) = a1logx1 +a2e

x2 + a3x1x2. Determine values of the parameters (if any) for which f is concave, for which f is

convex, and for which f is neither concave nor convex.

5. A critical point of a differentiable function f : Rn → R is a point x ∈ Rn at which ∇f(x) = 0— i.e., at which all partial derivatives have the value zero. For each of the following exercises

in Simon & Blume determine the critical points of the function and determine at each critical

point whether the Hessian matrix (the matrix of second partial derivatives) is positive or negative

definite or semidefinite or indefinite:

#17.1(a), 17.1(b), and 17.2(a).

The solutions in the back of the book for these three problems are not all correct.

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Math Camp 2020: Exercise Set #5

1. Define f : R2 → R by f(x1, x2) = x1x2(3 − x1 − x2). Note that f is twice differentiableeverywhere in R2. (Actually, as a polynomial, derivatives of f of all orders exist on all of R2.)

(a) Find all the critical points of f — i.e., all points x ∈ R2 at which ∇f(x) = 0.

(b) Determine the Hessian matrix D2f(x1, x2) as a function of (x1, x2).

(c) Determine all local maxima or local minima of f , if any.

(d) Determine all global maxima or minima of f , if any.

(e) At the point (x1, x2) = (1, 1) write the 2nd-degree Taylor polynomial in ∆x1 and ∆x2 — i.e.,

write all the polynomial’s terms.

2. Suppose that we have n data points (x1, y1), . . . , (xn, yn) and we want to determine the least-

squares line y = mx+ b for the given data. This is the line that minimizes the sum of the squares

of the n residuals ri = mxi + b− yi, i.e., that minimizes the function

S(m, b) =n∑

i=1

(mxi + b− yi)2.

Note that the xis and yis are not variables in this problem, they’re the data, which we’re given.

The variables are m and b, the slope and intercept of the line that minimizes the sum of squares.

(a) Draw a diagram depicting this problem with six data points. Indicate the residuals in the

diagram.

(b) Prove that the least-squares line has slope and intercept

m =n∑xiyi − (

∑xi)(

∑yi)

n∑x2i − (

∑xi)2

and b =(∑x2i )(

∑yi)− (

∑xi)(

∑xiyi)

n∑x2i − (

∑xi)2

.

3. (See Exercise #21.12 in S&B) Assume that a one-product firm faces an inverse demand

function p = F (q) for its product and that the firm’s cost function is C(q). Assume that F (·) andC(·) are both twice differentiable.

(a) Obtain expressions for the firm’s revenue function and its marginal revenue function.

(b) Determine some condition(s) that are sufficient to ensure that the firm’s profit function π(q)

is concave, and some condition(s) that guarantee π(·) is strictly concave, What are the conditions

for concavity and strict concavity of π when the demand function is linear? What would be the

significance of π(·) being strictly concave?

For the remainder of this exercise assume that F is the linear function F (q) = a−bq on the domain0 5 q 5 a/b; that C(q) = cq; and that a, b, c > 0 and a > c.

(c) Write down the firm’s revenue function and its profit function, and obtain its marginal revenue

and marginal cost functions. All of these functions should have one decision variable and three

parameters — for example, π(q; a, b, c).

(d) Obtain the solution function of the firm’s profit-maximization problem.

(e) Obtain the value function for the firm’s profit-maximization problem.

(f) Determine the derivative of the solution function with respect to the parameter c.

(g) Determine the derivative of the value function with respect to the parameter c. Compare this

to the derivative of the profit function with respect to c. How do you account for this comparison,

in light of the fact that the optimal level of output changes in response to changes in c?

4. Assume that a > 0 and b > 0. On the domain R2+ define the functions

f(x; a) = ax1 + x2, G1(x) = 2x1 + x2, G

2(x) = 2x1 + 4x2, G3(x) = x1 + x2

and the maximization problem

(P) max f(x; a) s.t. G1(x) 5 9, G2(x) 5 18, G3(x) 5 b.

For parts (a), (b), and (c) assume that a = 1, so the problem’s only parameter is b.

(a) For each of the following values of b, draw the constraint set (the feasible set): b = 3, b = 5,

and b = 8.

(b) For values of b greater than 8, determine the solution of the problem (P); draw the gradients of

the functions f , G1, and G2 at the solution; and determine the values of the Lagrange multipliers

at the solution — i.e., determine ∇f as a non-negative linear combination of the three G-gradients.Which constraints are binding at the solution and which are non-binding?

(c) Determine the solution function for the problem (P). Note that for some values of b the solution

function is not really a function: there are multiple solutions. Which values of b are those, and

what are the solutions (the maximizers of f) in those cases? Determine the value function for the

problem (P).

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For parts (d), (e), and (f) assume that b = 5, so the problem’s only parameter is a.

(d) Draw the feasible set. Including the non-negativity constraints, there are 10 intersections of

pairs of constraints. Identify which intersections are feasible points (vectors) and label them A, B,

C, etc. At each of these feasible, labeled intersections, identify which constraints are binding and

draw the gradients of the Gi functions for the binding “structural” constraints (i.e., the binding

Gi-constraints).

(e) At each intersection in part (d), determine the values of a for which that intersection is the

optimal solution. Determine the solution function for the problem (P). Note that for some values

of a the solution function is not really a function: there are multiple solutions. Which values of a

are those, and what are the solutions (the maximizers of f) in those cases? Determine the value

function for the problem (P) — remember, a is the only parameter in this part.

(f) For each intersection in part (d), determine the values of the Lagrange multipliers when that

intersection is an optimal solution — i.e., determine ∇f as a non-negative linear combination ofthe three G-gradients. The Lagrange multiplier values will depend upon a. (If you find it difficult

to do this part for arbitrary values of a, then assume successively that a = 1/4, a = 3/4, a = 3/2,

and a = 3.) At each intersection indicate the cone in which ∇f must lie.

5. Let f : R → R be a real function and let x ∈ R be a real number at which the first, second,and third derivatives of f all exist. Verify that the third-degree Taylor polynomial

P3(∆x) := f(x) + f′(x)∆x+

1

2f ′′(x)(∆x)2 +

1

6f ′′′(x)(∆x)3

that we use to approximate the function f near x has

(a) the same value as f at x (i.e., at ∆x = 0),

(b) the same slope (the same derivative) as f at x,

(c) the same curvature (the same second derivative) as f at x, and

(d) the same third derivative as f at x.

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Math Camp 2020: Exercise Set #6

1. The Kuhn-Tucker sufficient-condition theorem includes the following second-order conditions:

• each Gi is quasiconvex, and either

• f is concave, or

• f is quasiconcave and ∇f 6= 0 at x̂.

What we would like is a counterexample demonstrating that we can’t dispense with the condition

that ∇f 6= 0 at x̂ when f is merely quasiconcave but not concave. Provide such a counterexampleusing the following function from Exercise Set #3:

f(x1, x2) =

x1x2, if x1x2 ≤ 4

4, if 4 < x1x2 < 9

x1x2 − 5, if x1x2 ≥ 9

2. Define f : R2+ → R by f(x1, x2) = x1x2. We wish to maximize f subject to the constraintG(x1, x2) 5 c, where G(x1, x2) = x21 + x

22 and c > 0.

(a) Determine the gradients of f and G as functions of x = (x1, x2).

(b) Determine the solution function of the maximization problem.

(c) Draw a diagram depicting the constraint, the solution, the objective-function contour through

the solution, and the gradients of the objective and constraint functions at the solution.

(d) Determine the value function of the maximization problem.

(e) Verify that for every value of c the Kuhn-Tucker conditions are satisfied at the solution.

(f) Determine the value of the Lagrange multiplier at the solution.

(g) Assume that c = 2. What is the solution of the maximization problem? Apply the theorem

on the last page of the lecture notes “Second-Order Conditions and Quadratic Forms with Con-

straints” to verify that the sufficient conditions for a constrained maximum are satisfied at your

solution.

(h) On your diagram in (c), draw the line tangent to the constraint at the solution. Determine

whether the objective function is positive or negative definite or semidefinite on this line.

3. It’s trivial to show that the function f(x1, x2) = x1x2, defined on the domain R2++, is strictlyconvex if restricted to a line x2 = ax1 in R2++ for any a > 0. Use the theorem on page 2 of the“Differentiable Quasiconcave Functions” lecture notes to verify that f is nevertheless quasiconcave

on the entire domain R2++.

4. In order to apply the same theorem as in #3 to the function f(x1, x2, x3) = x1x2x3 defined on

the domain R3++, you will need to check the signs of certain determinants.

(a) Write down in detail each of those determinants.

(b) In lieu of you evaluating the determinant of the 4× 4 bordered matrix, I will tell you that thedeterminant is −3(x1x2x3)2. Any other determinants you will need for this question you shouldbe able to evaluate. Use the theorem to determine whether f is quasiconcave, quasiconvex, or

neither, by determining the signs of the required determinants.

5. In an example we determined the definiteness of the quadratic form f(x1, x2) = x21−x22 on any

line through the zero vector. Instead, use the theorem on page 8 of the lecture notes “Second-Order

Conditions and Quadratic Forms with Constraints” to obtain the same result, about whether f is

positive or negative definite or semidefinite, or indefinite, on any given line x2 = ax1.

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